C₆₁×#12381 #29515814

order	15616
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₆₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₆₁×#12381

Subgroups

	subgroup	G/H
center, socle	C₂²×C₆₁	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₆₁
prime radical	C₆₁×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

306 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 7808

A minimal faithful permutation representation — 3 generators on 93 points:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,23)(19,22)(20,27)(24,32)(25,31)(26,29)(28,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(4,13)(7,12)(9,16)(11,14)(17,18)(19,24)(20,28)(21,25)(22,30)(23,27)(26,31)(29,32);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,19,21,22)(18,24,25,30)(20,26,29,27)(23,28,31,32)

Conjugacy classes & character table

2440 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 5369);